Using Standard Methods

set.seed(123)
library(mgcv)
dat = gamSim(1, n=400, dist="normal", scale=1, verbose=F)
mod = lm(y ~ x1 + x2, data=dat)
summary(mod)

Call:
lm(formula = y ~ x1 + x2, data = dat)

Residuals:
    Min      1Q  Median      3Q     Max 
-8.6634 -1.3220 -0.0571  1.5297  7.0937 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept)   7.2745     0.3332   21.83   <2e-16 ***
x1            6.3616     0.4352   14.62   <2e-16 ***
x2           -5.1214     0.4526  -11.31   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 2.532 on 397 degrees of freedom
Multiple R-squared:  0.4594,    Adjusted R-squared:  0.4567 
F-statistic: 168.7 on 2 and 397 DF,  p-value: < 2.2e-16

# requires broom and glue packages
broom::augment(mod) %>% 
  ggplot(aes(x=.fitted, y=y)) +
  geom_point(alpha=.25, color='#ff5500') + 
  geom_smooth(se=F, color='#00aaff') +
  annotate('text',
           label = glue::glue("Rsq = {round(summary(mod)$r.squared, 2)}"),
           x = 4,
           y = 16) +
  labs(title='Fitted vs. Observed') +
  theme_minimal()

Heteroscedasticity, non-normality, etc.

modlog = lm(log(y) ~ x1 + x2, dat)
summary(modlog)
plot(modlog, which = 1:2)

broom::augment(mod) %>% 
  ggplot(aes(x=.fitted, y=y)) +
  geom_point(alpha=.25, color='#ff5500') + 
  geom_smooth(se=F, color='#00aaff') +
  annotate('text',
           label = glue::glue("Rsq = {round(summary(modlog)$r.squared, 2)}"),
           x = 4,
           y = 16) +
  labs(title='Fitted vs. Observed') +
  theme_minimal()

Polynomial Regression

set.seed(123)
x = runif(500)
mu = sin(2 * (4 * x - 2)) + 2 * exp(-(16 ^ 2) * ((x - .5) ^ 2))
y = rnorm(500, mu, .3)
d = data.frame(x,y)

Polynomial regression is problematic

A standard linear regression is definitely not going to capture this relationship. As above, we could try and use polynomial regression here, e.g. fitting a quadratic or cubic function within the standard regression framework. However, this is unrealistic at best and at worst isn’t useful for complex relationships. In the following, even with a polynomial of degree 15 the fit is fairly poor in many areas, and ‘wiggles’ in some places where there doesn’t appear to be a need to.

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